The articles and essays in this blog range from the short to the long. Many of the posts are also introductory (i.e., educational) in nature; though, even when introductory, they still include additional commentary. Older material (dating back mainly to 2005) is being added to this blog over time.

Friday, 27 February 2015

(A) The sentence A is not true (Part Two)

(A)
The sentence A is not true.

I've
never come across an argument against the lack of propositional
content position on (A). I've read references to (A)'s lack
of content – though that was it! The subject is quickly passed over
as if to say this:

The
logic and the paradoxes apply even if the sentence has no
propositional content.

However,
what if the logic and the paradoxes don't apply if the sentence has
no content? Or, to put that another way, perhaps the paradoxes only
arise because it has no propositional content. (Indeed
wouldn't this also apply to the Liar Paradox?)

I
understand that all the self-referential paradoxes work and how they
come about. However, that may only be because the lack of content is
ignored (or dismissed). So, sure, it works fantastically well as
logical puzzle if the lack of content is ignored. The problem is: Can
that the lack of content be ignored?

I'm
also having a problem tying my criticisms in with the philosophical
and historical fact that these self-referential paradoxes have had a
profound effect on logic, mathematics and philosophy.

So
perhaps it's all down to syntax and not semantics. That is, it's
about the form/syntax of the sentences (as well as the
problems/puzzles/paradoxes): not the content. Though if that's true,
isn't it a sleight of hand to use sentences which appear to
have content?

If my
quibbles about propositional content are irrelevant, I will still need to
know why that's so. If it's all down to logic and syntax (or form) ,
then sentences like “Everything I say is a lie” or “(A) This
sentence A is not true” are misleading.

And if
it's all about p's, q's, entailment, etc., and not
really about propositional content, then why use sentences like “What
I am saying is a lie” or “(A) This sentence A is not true” at all?
The pretence or appearance of having propositional content is either
not needed at all or just a plain mistake. If it's not about
propositional content, logicians and philosophers shouldn't use
examples which appear to have propositional content.

However,
despite not finding much in the philosophical literature, one person
did write this: “The liar isn't about propositional content.”

Perhaps
not. Though what about (A)? And why isn't the Liar Paradox also about
content? Or, at the very least, why isn't content relevant? That's
precisely the question.

Indeed
the person who said that content is not required then went on to talk
about the Liar Paradox. He said:

“It
led to the collapse of logicism and indirectly to Gödel's
incompleteness results (i.e., that in a formal system like
Zermelo-Frankel set theory you can derive (G(F) = "This sentence
cannot be proved in F".)”

Yet
that's just history and context.

In any
case, Gödel's “This sentence cannot be proved in F” doesn't
seem like (A) or the Liar Paradox – at least it's not precisely the
same.

Put it
this way.

Sentence
S in system X is true though it can't be proven to be true in X.

isn't
like a sentence which refers to itself – as in (A). It's saying
that a statement or mathematical truth within a system is true even
though it can't be proven within that system. The problem with (A) or
the Liar Paradox is that it can't be established if it's true or
false (or it's both). Gödel's 'S in X' is true – just not proven
to be true within the system to which it belongs.

Propositional
Content & Plain Content

It's
not being argued here that sentences have to be truth-evaluable in
order to be legitimate. Indeed a distinction should be made between
propositional content and mere or plain content.

A
sentence with propositional content is truth-evaluable (it can
be true or false); whereas a sentence with only content need
not be true or false (though it still has content).

For
example, the sentence “Shut that door” has content; though not
propositional content. It's a command. (In fact it has more content,
as it were, than “I'm lying at this moment”.)

The
bottom line is not that all these sentences or statements must have
propositional content (or be capable of being true or false). The
argument is that only (A) and the Liar Paradox must have propositional
content in order to be true or false (at least in principle). If they
aren't propositional in that sense, then they can't generate the
paradoxes.

A
sentence can be self-referential and also have semantic (or
otherwise) content. It's just that (A) doesn't.

We
could have:

“When
I say that 'All cows have four legs is true', what I'm saying is
false.”

Of
course that doesn't really work. One, it's really more than one
sentence. Two, it's a plain self-contradiction. On the other hand,
there's nothing self-contradictory in:

(A)
The sentence A is not true.

Despite
all that, some have commented on the non-problematic nature of
self-referential statements. However, I have no problem with all
self-referential statements; just some of them: the ones without
propositional content and which constitute the self-referential
paradoxes.

For
example, someone gave this example:

“I
am lying to you at this very moment.”

He
then went on to say that "no one can object that that sentence has
no content”.

Grammatically
speaking, the sentence “I am lying to you at this very moment” is
a super sentence. We all know what the individual words means and it
- grammatically - seems to make sense. However, what is its
propositional content?

It can
have propositional content if the self-accusation of lying refers to
other statements the speaker has made. (Those other sentences would
then be false.) However, it's supposed to be a self-referential
statement. What is X lying about? He can't be referring to his lying
alone because in order to lie, you have to make a claim that is false
and to know that it's false.

The
fact is that he isn't lying or telling the truth. X is only making a
grammatically- acceptable sentence which has no propositional content.
Therefore he can't be lying.

(You
can also ask this question: If S has no content as such, then why is
it grammatically acceptable?)

So
compare

“I
am lying to you at this very moment.”

with

“I
am singing to you at this very moment.”

The
two aren't equivalent and not just because one is about lying and the
other is about singing.

When
someone says “I am singing to you at this very moment” he is
either lying or telling the truth. (He could be singing those words.)
That doesn't also work for “I am lying to you at this very moment”.
It has the same grammatical form; though the sentences differ not
just in content. The latter is neither true nor false. The former is
either true or false. And even if they have the same grammatical
form, one is has a truth-value and the other doesn't. (Indeed I would
argue that because of this difference, it surely can't even be said
that they have the same grammatical form.)

Again,
because “I'm singing to you at this very moment” and “I'm lying
to you at this very moment” have the same shape (or form), this
creates the problems. They may well have the same grammatical shape.
Though one could be true and the other is neither true nor false.
That difference seems to be very clear.

The
Barber Paradox

It's
worth noting here that some of the other well-known paradoxes are of
a different nature to (A) and the Liar Paradox.

For
example, take the Barber Paradox.

The
Barber Paradox isn't about a single sentence. It can only be
established through a chain of arguments. It's self-referential in
that it deals with the question of whether the barber does or doesn't
shave himself; though it's not about a self-referential statement
(or sentence) as in the Liar Paradox or (A).

Of
course the Barber Paradox can be summed up in a single sentence, such
as:

“I
shave everyone who doesn’t shave themselves.”

However,
it's still not a self-referential sentence like “I am lying at this
very moment”. It's about a self-referential situation (as it
were); though, unlike (A) and the Lair Paradox, it isn't about a
single sentence referring to itself or a person referring to what
he's currently saying. The sentence above sums up the situation. It's
about a possible or impossible state of affairs; not about a
self-referential statement.